Abstract
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adolphson, A.: Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), 269-290.
Batyrev, V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535.
Cattani, E., Cox, D. and Dickenstein, A.: Residues in toric varieties, Compositio Math. 108 (1997), 35-76.
Cattani, E., D'Andrea, C. and Dickenstein, A.: The \({\mathcal{A}}\)-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), 179-207.
Cattani, E. and Dickenstein, A.: A global view of residues in the torus, J. Pure Appl. Algebra 117/118 (1997), 119-144.
Cattani, E., Dickenstein, A. and Sturmfels, B.: Residues and resultants, J. Math. Sci. Univ. Tokyo 5 (1998), 119-148.
Cattani, E., Dickenstein, A. and Sturmfels, B.: Binomial residues, Submitted, http://anXiv.org/abs/math.A6/0003178.
Cox, D.: Toric residues, Arkiv Mat. 34 (1996), 73-96.
Gel'fand, I. M., Zelevinsky, A. and Kapranov, M.: Hypergeometric functions and toral manifolds, Funct. Anal. Appl. 23 (1989), 94-106.
Gel'fand, I. M., Kapranov, M. and Zelevinsky, A.: Generalized Euler integrals and Λ-hypergeometric functions, Adv. in Math. 84 (1990), 255-271.
Gel'fand, I. M., Kapranov, M. and Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, 1994.
Kapranov, M. M., Sturmfels, B. and Zelevinsky, A. V.: Chow polytopes and general resultants, Duke Math. J. 67 (1992), 189-218.
Khovanskii, A. G.: Newton polyhedra and the genus of complete intersections, Funct. Anal. Appl. 12 (1978), 51-61.
Matsumura, H.: Commutative Ring Theory, Cambridge Univ. Press, 1986.
Pedersen, P. and Sturmfels, B.: Product formulas for resultants and Chow forms, Math. Zeit. 214 (1993), 377-396.
van der Put, M. and Singer, M. F.: Galois Theory of Difference Equations, Lecture Notes in Math. 1666, Springer, New York, 1997.
Sturmfels, B.: Gröbner Bases and Convex Polytopes, Amer. Math. Soc., Providence, 1995.
Saito, M., Sturmfels, B. and Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Math. 6, Springer-Verlag, Heidelberg, 1999.
Stanley, R.: Enumerative Combinatorics, Volume I, Cambridge Stud. Adv. Math. 49, Cambridge Univ. Press, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cattani, E., Dickenstein, A. & Sturmfels, B. Rational Hypergeometric Functions. Compositio Mathematica 128, 217–240 (2001). https://doi.org/10.1023/A:1017541231618
Issue Date:
DOI: https://doi.org/10.1023/A:1017541231618